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# If $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are three coplanar vectors then, $[(2\overrightarrow a-\overrightarrow b)\:(2\overrightarrow b-\overrightarrow c)\:(2\overrightarrow c-\overrightarrow a)]=?$

$(a)\:0\:\:\:\qquad\:\:(b)\:\:1\:\:\:\qquad\:\:(c)\:\:\sqrt 3\:\:\:\qquad\:\:(d)\:\:None\:of\:these.$
Can you answer this question?

Since $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are coplanar, $[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=0$
Now
$[(2\overrightarrow a-\overrightarrow b)\:(2\overrightarrow b -\overrightarrow c)\:(2\overrightarrow c-\overrightarrow a)]=\left |\begin {array}{ccc} 2 & -1 & 0 \\0 & 2 & -1 \\ -1 & 0 & 2\end {array}\right |\:[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]$
$=7\:[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=0$
answered Jan 5, 2014